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Jun 24, 2014 at 14:28 comment added Per Alexandersson Yes, see my comment in the end... i misread the question..
Jun 24, 2014 at 12:51 comment added Robert Bryant It is not true that the eigenvalues of $S$ have to be nonnegative and real, even in the case $n=2$. As shown in my answer, $S=-I_2$ has solutions, in fact, a $1$-parameter family of them.
Dec 23, 2013 at 21:14 history answered Per Alexandersson CC BY-SA 3.0